Flow Generated by a Suddenly Heated Flat Plate

Abstract

By employing the two-sided Maxwellian in Maxwell's moment method a kinetic theory description is obtained of the flow generated by a step-function increase in the temperature of an infinite flat plate. Four moments are employed in order to satisfy the three conservation equations, plus one additional equation involving the heat flux in the direction normal to the plate. For a small temperature rise the equations are linearized, and closed-form solutions are obtained for small and large time in terms of the average collision time. Initially the disturbances propagate along two distinct characteristics, but the discontinuities across these waves damp out as time increases. At large time the main disturbance propagates with the isentropic sound speed. Solutions for mean normal velocity and temperature show the transition from the nearly collision-free regime to the Navier-Stokes-Fourier regime, which is characterized by a boundary layer near the plate surface merging into a diffuse "wave". The classical continuum equations, plus a temperature jump boundary condition, seem to be perfectly adequate to describe the flow beyond a few collision times, provided one accounts properly for the interaction between the inner thermal layer and the outer diffuse wave.